arXiv subjects

Cunxiang Duan

Publications and source records attributed to Cunxiang Duan.

6 recordsLinked to original sources

arXiv2026

Spectral Extremal Graphs without a $K_k$-Factor

Let $k\ge 3$ and let $n=km$. A $K_k$-factor in an $n$-vertex graph is a collection of $m$ vertex-disjoint copies of $K_k$ that covers the entire vertex set. We determine the maximum adjacency spectral radius of an $n$-vertex graph containing no $K_k$-factor when $m\ge 2k-1$. More precisely, we prove that every such graph $G$ satisfies \[ ρ(G)\le ρ(H_{n,k}), \qquad H_{n,k}=K_{k-2}\vee\bigl(K_{n-k+1}\cup K_1\bigr), \] with equality if and only if $G\cong H_{n,k}$. Equivalently, the unique extremal graph is obtained from $K_{n-1}$ by adding one vertex adjacent to exactly $k-2$ vertices of the clique. Our proof combines a decomposition lemma for sparse complements, derived from the Hajnal--Szemerédi theorem, with the Motzkin--Straus inequality and spectral estimates based on quotient matrices and the Rayleigh quotient.

math.CO↗

arXiv2023

The characteristic polynomials of uniform hypercycles with length four

Let $C_{m}$ be a cycle with length $m.$ The $k$-uniform hypercycle with length $m$ obtained by adding $k-2$ new vertices in every edge of $C_{m},$ denoted by $C_{m,k}.$ In this paper, we obtain some trace formulas of uniform hypercycles with length four. Moreover, we give the characteristic polynomials of uniform hypercycles with length four.

math.SP↗

arXiv2022

Signless Laplacian Estrada index and Laplacian Estrada index of uniform hypergraphs

We generalize the notions of Laplacian and signless Laplacian Estrada index to uniform hypergraphs. For an $r$-uniform hypergraph $H,$ we derive an order $r+1$ trace formula of the (signless) Laplacian tensor of $H.$ Among others by using this trace formula, we obtain lower bounds for the signless Laplacian Estrada index and upper bounds for the Laplacian Estrada index. Moreover, we establish a bound involving both the Laplacian Estrada index and Laplacian energy of a uniform hypergraph.

math.CO↗

arXiv2020

The bounds of the spectral radius of general hypergraphs in terms of clique number

The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.

math.CO↗

arXiv2018

The largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

Let $S_{1}(m, d, k)$ be the $k$-uniform supertree obtained from a loose path $P:v_{1}, e_{1}, v_{2}, \ldots,v_{d}, e_{d}, v_{d+1}$ with length $d$ by attaching $m-d$ edges at vertex $v_{\lfloor\frac{d}{2}\rfloor+1}.$ Let $\mathbb{S}(m,d,k)$ be the set of $k$-uniform supertrees with $m$ edges and diameter $d$ and $q(G)$ be the signless Laplacian spectral radius of a $k$-uniform hypergraph $G$. In this paper, we mainly determine $S_{1}(m,d,k)$ with the largest signless Laplacian spectral radius among all supertrees in $\mathbb{S}(m,d,k)$ for $3\leq d\leq m-1$. Furthermore, we determine the unique uniform supertree with the maximum signless Laplacian spectral radius among all the uniform supertrees with $n$ vertices and pendent edges (vertices).

math.CO↗

arXiv2018

Edge Connectivity, Packing Spanning Trees, and Eigenvalues of Graphs

Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup V_{2})\neq ϕ$ and edge connectivity $κ'(G)=e(V_{i},V(G)\backslash V_{i})$ for $1\leq i \leq 2$. A multigraph is a graph with possible multiple edges, but no loops. Let $τ(G)$ be the maximum number of edge-disjoint spanning trees of a graph $G$. Motivated by a question of Seymour on the relationship between eigenvalues of a graph $G$ and bounds of $τ(G)$, we mainly give the relationship between the third largest (signless Laplacian) eigenvalue and the bound of $κ'(G)$ and $τ(G)$ of a simple graph or a multigraph $G\in\mathcal{G}$, respectively.

math.CO↗